Researching differences between reform and traditional instructional methods in mathematics (collaborative post) The difference between traditional and reform instructional practices needs to begin with a history lesson. Our country has attempted to reform math instruction since after World War II. There were concerns the United States was lagging behind in technology due to the Russian launch of the satellite Sputnik. The space race had begun. However, prior to this launch the University of Illinois Committee on School Mathematics was already investigating problems with how high school mathematics was being taught. There were other influences involved in this “New Math” such as Piaget’s stages of cognitive development, Bruner’s discovery of mathematical ideas and his three stages of representation of mathematical ideas. New Math was in full swing in the 1960’s, however; in the 1970’s it all disappeared and teaching returned to “back to basics”. (Herrera,Owens, 2001, p.85) The NCTM Standards-Based Reform approach to teaching mathematics was launched in 1989 after results from two international studies were published. They were the Second International Mathematics Study and the International Assessment of Educational Progress. Both studies demonstrated a decline or a stand-still of math scores in the United States. The National Commission for Excellence in Education published A Nation at Risk in 1983 sparking a sense of urgency with its quote, “If an unfriendly foreign power had attempted to impose on America the mediocre educational performance that exists today, we might have viewed it as an act of war.” (NCEE, 1983, p.5) There was also an update in 2000 Principles and Standards for School Mathematics. The six principles of fundamental to high quality mathematics instruction from the Principles and Standards for School Mathematics are:
The Equity Principle
The Curriculum Principle
The Teaching Principle
The Learning Principle
The Assessment Principle
The Technology Principle
A traditional mathematics curriculum is based on direct instruction, such as, lecture style teaching, algorithms, and memorization. While a reform, or standards-based curriculum, is based on reasoning and construction through problem solving and investigation. (Baker, 2010) For Example, in a ttraditional mathematics curriculum, a student needs to know 3x5=15 before the student goes on to solve this word problem: Jones bought 3 different color pairs of socks and five different kinds of blue jeans. How many new combinations of socks and jeans can Jones make? (Latterell and Copes, 2003) Conversely, in a rreform mathematics curriculum, students learn the basics at the same time while solving far more practicle problems, so that students will be prepared to solve problems in real world situations. (Latterell and Copes, 2003) In tradition mathematics settings the students learn the basics so that they can explore and conquer more complex problems. In all aspects of learning you have to learn the basics before you can proceed to something harder. Critics may say that in some cases traditional instruction is less effective than other alternative methods of teaching but in some instances its the best for certain students/ group of students. Each part of the country/demographic area is different when it comes to learning. One area may learn best with traditional math instruction, where the school down the road may find a new mathematical method more effective. Reform mathematics curriculum challenges student to make sense of new mathematical ideas through exploration and projects. Not by filling in a formula or doing the question/problem a certain way. Reform text books emphasize written and verbal communication, group work, making connections between concepts and representations, unlike traditional texts which emphasize step by step processes of problems with examples. New Math has an emphasis on “ deductive reasoning, set theory, rigorous proof, and abstraction, while the Standards emphasize applications in the real world context, especially experimentation and data analysis.” (Herrera, Owens, (2001), p. 91) Critics of the standards-based approach suggest that reform mathematics puts too much emphasis on the effort, and not on the solution; educators are praising the process, regardless of the product. However, creators of standards-based curriculums argue that the specific content has not changed, “rather it shows how reasoning and sense making can be incorporated throughout the curriculum.” (Baker, 2010) The Standards-Based Reform has been slow to take off. Classes are still mostly lecture style with little conversation between teacher and students. Are we stubborn and do not want to give up our “right” or “wrong” approach to teaching and learning math? (Newton, 2007, p. 6) Or does government place too many demands on teachers and schools to implement the reform? With the math reform, many new aspects of teaching math have came out. Instead of having students just sit and listen to a lecture during class, more group work and hands on studies are being introduced. At the school that I work at, we have started a new program this year, called Connected Math. This is a very hands on lesson and has more group work than individual work. With this program students are suppose to learn from each other instead of just the teacher. Van de Walle and Lovin both describe that group work is a better way for students to learn becasue they are learing from their peers instead or a superior in Teaching Student-Centered Mathematics. References: Baker, Linda. Numbers Wars: School Battles Heat Up Again in the Traditional versus Reform-Math Debate. Scientific American, March2010, 2p, 1 Illustration Herrara, Terese A., Owens, Douglas T. The “New New Math”?: Two Reform Movements in Mathematical Education. Theory Into Practice, Spring2001, Vol. 40 Issue 2, p84, 9p
Latterell, Carman M., Copes Lawrence. "Can We Research Conclusion in Mathematics Education Research?. Phi Delta Kappani; Nov2003, Vol. 85 Issue 3, p207-211, 5p, 1 Illustration Newton, Xiaoxia. Reflections on Math Reforms in the U.S.: A Cross-National Perspective. Education Digest, Sep2007, Vol. 73 Issue 1, p4-9, 6p
Lovin, LouAnn H., Van de Walle, John A. "Teaching Student-Centered Mathematics Grades 5-8", 2006, Vol.3, pg 1-37 Integration of Reform vs. Traditional Instructional Practices “ Ten years after NCTM’s release of the Curriculum and Evaluation Standards for School Mathematics, this country is having a wrenching debate about what should be taught in mathematics and how it should be taught. Debate has degenerated to “math wars.” On one side are those who fervently believe children need to learn “the basics.” On the other side are those who believe or think they believe in the message of the Standards. These are the educators who believe in “reform mathematics.” Objectivity often gets lost in rhetoric and what either side believes is frequently vague at best (Van de Walle, 2003).” Briefly, some advantages and disadvantages of each philosophy are listed below:
Advantages
Disadvantages
Traditional
Reform
Traditional
Reform
Basics are stressed heavily
Connections made more easily
Concepts appear unrelated
Basics not stressed
Most textbooks are in a more traditional format
Focus on particular areas to gain depth of understanding
Textbook publishers include too much info therefore creating many 1 day lessons
Textbook must be filtered to determine what is really important
Ability to keep on schedule with a “get it or not” philosophy
Inquiry oriented approach
Learn by rote does not allow students time to compartmentalize
Time constraints based on standardized curriculum
Provides avenue to learn the basics as a building block
Emphasis on conceptual understanding
No depth of understanding
Surface level understanding of mathematics as a rule
While both methods of instruction have issues to contend with, they both also have advantages that could be incorporated with one another for a more holistic approach. In an interview with the Baltimore Curriculum Project, University of MD Mathematics Professor Jerome Dancis stated “ In fact, mathematicians advocate the importance of both conceptual understanding (Reform) and basic skills (Traditional). They are not mutually exclusive. Basic skills are necessary for conceptual understanding and problem-solving ("Interview with university," 2006).”o < /span> Would it not be possible then to instill the basics in students while actively seeking to utilize the concepts developed by the NCTM? In the text “Elementary and Middle School Mathematics: Teaching Developmentally (Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M., 2010, p. 5).”t t the authors refer to a document (The Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (2006)) created by experts who were brought together by the NCTM. Their thoughts on coherence basically suggest that the most significant concepts and skills be emphasized in each grade so as to form a fundamental core content for each grade. With a more coherent structure in place that is built upon year after year, the teacher will have more time to focus on learning methods and spend less time attempting to connect the student’s present learning with learning from the past. The students will have a much better chance to make “connections” on their own. However, this is only one example of an approach that could be employed to integrate the two philosophies. In order to effectively integrate the two will require much work on the part of the teacher. The traditional methods will need to be studied, along with the reforms that have been suggested in an effort to align the two. It might be that this approach will need to be addressed with each individual lesson or concept. But, in order to address the most critical issue, the student’s education, such efforts are a necessity. References Van de Walle, J.A. (2003, April 1). Reform mathematics vs. the basics: understanding the conflict and dealing with it. Retrieved from http://mathematicallysane.com/reform-mathematics-vs-the-basics/ Interview with university of md mathematics professor jerome dancis (2006, February). [Electronic mailing list message]. Retrieved from http://www.baltimorecp.org/newsletter/BCPnews_feb06.htm#spotlight Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2010). Elementary and middle school mathematics: teaching developmentally. Allyn & Bacon. The comparison between traditional and reform practices in mathematics is marked by a healthy debate on the advantages of each educational approach. The authors of Elementary and Middle School Mathematics, point out that although empirical data regarding the effectiveness of each approach is still being gather, it appears clear that students in standard-based programs outperform traditional students in .problem-solving measures ( Van De Valle, Karp, Bay-Williams, 2010). This page presents a collection of studies relevant to Mathematics reform.
Learning From Teaching: Exploring the Relationship Between Reform Curriculum and Equity. Jo Boaler
Journal for Research in Mathematics Education. July 2002, Volume 33, Issue 4, Pages 239 - 258
The author discuses two studies which suggest that reform-based teaching and learning are central to the attainment of linguistic, ethnic, and class equity in schools.
Problem Solving as a Means Toward Mathematics for All: An Exploratory Look Through a Class Lens
Sarah Theule Lubienski
Journal for Research in Mathematics Education. July 2000, Volume 31, Issue 4, Pages 454 - 482
Leubienski studied the experiences of 7th grade with problem-solving approaches to mathematics pedagogy.
This brief from the National Council of Teachers of Mathematics analyzes the difficulty in producing conclusive comparative studies between traditional and standard-based methods of instruction. It does however suggests that the patterns of evidence indicate that while the both approaches appear to produce similar results in standardized computational tests, students taught with a standards-based approach seem "outperform students taught with traditional methods. Back to the top
Classroom Participation (collaborative observations from the EDMG 519 video "The Border Problem) Role of the teacher: · Facilitator · Encourage different methods of solving the problem. · Encourage students to demonstrate and explain to the rest of the class how they solved the problem. · Encourage making connections between the different methods. · Challenge the students to use one of the methods on a new problem.
Provide feedback and praise.
Bring into the discussion relevant information not brought up by the students.
The teacher wrote down the different methods on the chalk board in expression format.
Challange the students to find the last missing expressions from the board.
Challange the students to explain not only how but why they used the chosen method.
Maintain focus and interest throughout discussion.
v Thoroughly understand the material. v Elicit strategies from the students. v Discuss the various strategies. v Create a spirit of inquiry, trust, and expectation. v Make lessons relevant to real world applications. v Demonstrate various approaches of problem solving. v Involve technology in the classroom. v Elicit deeper thought as to method used. v Guide discussions. v Show respect for the students. v Ask contemplative questions of the students.
Challenge students/make them think/use what they know to solve abstract problems
Make sure students understand material that is being taught
Let students take turns answering and explaining their answers.
Writing down notes/equations students come up with to solve specific problems
1. Have tools and manipulatives that put math into world context, so they can see how it will affect them. 2. Earn the respect of the students by showing them that they care about their education, and by having the materials ready for the students. 3. Be open minded to new ideas, from other teachers and students. 4. Be constantly advancing in new technology in the class room.
Role of the students: · Explain to the rest of the class how they solved the problem correctly. · Explain to the rest of the class how they solved the problem incorrectly. · Make sure they understand how the other students solved the problem correctly. · Raise their hand quietly and wait to be called on (perhaps this one belongs in norms?)
work productively with the member in their group
participate in class discussion/ be ready to answer questions
Listen to others
Ask questions if you don't understand material
Respect classmates and their thoughts.
Participate in the discussion.
Think of their own solution/opinion.
Work in cooperation and community with others.
v Actively participate in classroom activities. v Learn from the methods of others within the class. v Listen to others without them fearing other student’s derision. v Work with others while demonstrating respect. v Exhibit respect for the teacher. v Learn. 1. Pay attention during class to the teacher and work. 2. Respect other student and teachers. 3. Keep up with text books, calculator, and notes from the class. 4. Contribute to the class every time therer is a discussion time. 5. Help other students understand what their method was in working a problem. 6. Have, learn proper knowledge of terms and wordings so explanations can be understood by entire class. Norms of the classroom: · Raise your hand to respond to the teacher’s request. · Do not distract from the discussion. Students know that when going to explain something in front of the class a pointer may be helpful. Students showed that they needed to raise their hand and put it down if they were not called upon. Students sat in their desks the right way and knew where they were suppose to sit. Student's new to give their attention to whoever was speaking whether it was the teacher or a fellow classmate.
Time management and control
Participate in the discussion without interupting teacher or other students.
Wait 3 minutes before raising hand to allow time for everyone to think about the question.
Everyone must listen attentively and quietly when someone else is speaking.
Faster and slower, more confident and less confident, students in the classroom.
v Teacher creates a spirit of inquiry, trust, and expectation. v Teacher involves technology in the classroom. v Teacher guides discussions. v Teacher shows respect for the students. v Teacher asks contemplative questions of the students. v Students actively participate in classroom activities. v Students learn from the methods of others within the class. v Students listen to others without them fearing other student’s derision. v Students work with others while demonstrating respect. v Students exhibit respect for the teacher.
Raise hand before speaking or if help is needed.
Have in depth discussion everyday about the methods and formulas that are taught that day
Have students as interactive in class as possible.
Use manipulatives as much as possible
Have real world examples for as much math as possible for the lesson.
Assessment Instruments
I agree that the P-PT tests skills rather than critical thinking. Also, the test writer made a few mistakes with the notation of a few of the problems (#20, 21, 23, & 24). The correct answer is not among the choices. For example #24; the question askes for 1.05 divided by 54.5. This yields 0.019 which is not among the choices. However, 54.5 divided by 1.05 yields 51.9, which is among the choices. (WB)
The pre-post test assesses basic number skills taught as "rules" in the math class.
Joyce, I agree[rfc]
I agree with Joyce that the pre-post test was seeing if the students knew the rules. I felt this test was testing students in standardize testing format ( multiple choice questions). (JT) I agree. (DB)
I total agree with you on that (Mitchell Reedy)
All of the questions on the Pre-Post Test are algorithmic, and involve decimals. There are no word problems or real-world examples except for the bonus question. The test is purely multiple choice, and the order of the questions suggests a specific sequence of skills. Often there are multiple questions testing a single skill presented in only one way. (EN)
The NAEP Item test assesses critical thinking skills learned in math class.
Joyce, I agree[rfc]
I agree with Joyce and I would use this test over the pre-post test because the critical thinking it involves. (JT)
I agree with JT & Joyce. (DB)
Our school had to take the NAEP test this year, and it was amazing how the calculator was not allowed in all parts, and that not all parts were multiple choice and they had to write out there answers and show all their work.
The NAEP test covers a variety of topics and skills, and each question is presented in a different way, even when accessing similar skills. There is a combination of multiple choice, short answer, discussion, and yes/no questions. Several of the questions could have more than one answer, or there are numerous pathways or methods that could be used to arrive at a solution. The test asks "Did you use a calculator?" after a few of the questions, implying that students may choose whether or not they would like to use one. (EN)
I like the how the NAEP is a variety of skills & topics and that the students really have to think about the question being asked instead of having just multiple choice. I think the students get to really show what they know with this test compare to P-PT. (DB)
Perhaps my strongest first impression was a huge contrast in the way I felt taking these tests. The P-PT immediately felt that it was difficult, tedious, and meaningless (just a bunch of annoying numbers); and I felt somewhat insecure about some of my answers even though I know the subject matter. In short: it was no fun at all!. In contrast, the NAEP felt like more fun, I felt more confident that even if I didn't know the answers or methods I could figure them out, and it was actually fun and challenging figuring them out. I felt more confident in my answers because the were relevant and make sense with respect to the evidence and clues I had in front of me.
What types of information do I gain from each?
I also agree. The NAEP test requires the students to understand the concepts involved and perhaps to assess how well they can visualize a problem. Also, this test requires the students to think more abstractly rather than just manipulating numbers. (WB)
PPT- The students' abilities to follow the basic rules of arithmetic.
Joyce, I agree[rfc]
The Pre-Post Test demonstrates the student's ability to reproduce algorithms previously taught in class with or without understanding, and with only short-term memorization. (EN
Will you truly know if your students can follow the rules of arithmetic from this test? I wouldn't know if my student knew the rules base on a multiple choice questions test. (JT)
I agree with you JT. You cannot really know what your students know by the P-PT. I think NAEP is a better way to assess that. (DB)
NAEP- The students' abilities to critically think about fraction and their relationships to one another.
Joyce, I agree[rfc]
I agree (Mitchell Reedy)
I agree with Joyce (JT)
The NAEP Test requires more critical thinking, and accesses a student's ability and understanding in applying previously taught skills to real-world situations. A students capability to solve a problem based on given examples is also tested. (EN) I agree (DB)
In addition of the good points already made, I think that as a teacher I gain a better insight into how well students understand the underlying ideas of the meaning of mathematics, of "doing mathematics"(De Valle, 2010). Also, I think the students themselves gain further understanding of the concept s being tested because they have to use their problem solving and critical thinking skills in completing the tasks, I am sure they would find many 'eureka" moments during the assesment that they might not have found before.
How might these items be an indicator of mathematics instruction in this classroom?
I think these tests will accurately reflect the type of instruction that the students receive on a daily basis. With the P-PT the students likely sit and work through problems out of te text each day without really understanding why they are doing this and the implications of the knowledge. The NAEP class will be more likely to have discussions concerning the concepts involved and will be required to work through real world applications that lead them to the fundamentals as a method of solving the problem. (WB)
I completely agree with you WB. Students need to understand the concepts they are learning so they can use them in everyday life and not just on tests.(DB)
PPT- More structured- straight from the textbook.
Joyce, I agree[rfc]
teaching towards a Standardize test (JT)
The Pre-Post Test class most likely uses lower-level demands, with explicit instruction techniques and no connection to underlying concepts. The focus is more on reproduction and correct answers than on understanding and cognitive development. (EN)
NAEP- Appropriate manipulatives were used and skills of drawing representations of the fractions were taught.
Joyce, I agree I would say however that it appears that skills of drawing representations were taught.[rfc]
My belief is the teacher will be having a discussion during the lesson with student's' to engage them in there learning. I also think the students will be working in groups to work with appropriate manipulatives were used and drawing representations of the problem to help solve the problem. (JT)
The NAEP Test class most likely uses higher level demands including procedures with connections and doing math. Skills are presented in a variety of ways, with the focus on developing a deeper understanding and either developing procedures, or applying previously learned material to real-world examples. (EN)
Yes, and I think the teaching strategy goes even beyond that to create in the students the ability and confidence that enables them to "do math". The NAEP test certainly seems to have built-in the expectation that students can use basic principles, together with their thinking and learning skills, to figure things out without the need of memorized algorithms if necessary.
I have another point here. If pre and post test activities from one instrument are desired, I think the PPT model will do a better job. Otherwise the NAEP test wi;ll do a better job of reflecting the student's abilities. [rfc]
My initial impression of the two assessments is that they are both totally different. The NAEP uses nothing but fractions for their questions, and the Pre-Post Test uses all decimals. The NAEP test also hits many different topics and the Pre-Post Test was addition, subtracting, and rounding. The NAEP test pushed for students to be able to work problems by hand instead of using the calculator for everything. The Pre-Post test didn’t push for one way or another, the use of calculator or not. They both are good tools because you get to see examples of what might be on the EOG’s, and you have the opportunity to teach students the importance of being able to do math without a calculator. With the Pre-Post Test, you get a sense of importance of estimating, which can sometimes can be over looked since students do it from 3rd grade up. Also a good number sense will help students with estimations, and being able to work without a calculator. These to samples, give a good insight as to some material that students will possible see again on a bigger test. Also it made me want to stress that students need to try to work problems without a calculator before they put it in there calculator. This can be an indicator of what happens in the classroom, just by the fact that one test says, no calculator, and the other doesn’t. One class has to use paper and pencil to solve problems. The other class seems like they might be more relaxed and just have to get close to the answer since it was more estimations. Both are good things to work on in a class, but they both need to be worked on.
The examples provided in the article Building Conceptual Bridges by Joseph G. R. Martinez are useful in helping middle school students transition from math to algebra.
2 + 2 + 2 = 6 or 3(2) a + a + a = 3a or 3(a)
22 ⋅ 23 = 32 or 25 a2 ⋅ a3 = a5
When introducing two step equations I have used candy to represent the variables and the constant.
I used Jolly Ranchers to show the variables and Hersey Kisses to show the constant. This was to show them they cannot mix the two.
I will use ∞ for Jolly Ranchers and Δ for Hersey Kisses .
3x + 2 = 8
∞∞∞ + ΔΔ = ΔΔΔΔΔΔΔΔ -ΔΔ -ΔΔ
∞∞∞ = ΔΔΔΔΔΔ
We can only subtract Jolly Ranchers from Jolly Ranchers in this problem. The candy represents the two separate parts of the equation and they cannot simply mix the two.
This gets the students to a one step equation they are more familiar with. This exercise and the related articles reiterate the fact that perhaps the best way to reach students with new mathematic concepts is to place the lessons into a format that holds some interest for them. The ideas of the lemonade mix, and also the dietary concerns of adolescents, both hit this target. Of course the problem of the lemonade mix is an older, more tried and true implementation of contextualizing to gain students interest. However, with the dietary concern project, there are several aspects that should be discussed. First would be the use of the media to find topics and establish the parameters for projects. Dietary concern seems to be a great application that will pique the interest of the students. While this is not a topic I would have used 30 years ago, by keeping current with the students interests will lead to such topics. As stated in Diet, Ratios, Proportions: A Healthy Mix “the importance of choosing a context should not be overlooked” (p. 13). Also, this is a great way to enter into conversations and learning concerning ratios. I have always found the use of ratios to be a very simple and effective way to answer the question “why do I need to know algebra?” Collectables, gas money, allowances, job opportunities are just a few. Furthermore, through this type of instruction, the implementation of charts and graphs becomes easy, almost second nature. And, by setting this up as a longer term project it will give the students ample opportunity to learn and continue to delve deeper into this interdisciplinary topic. The Third International Mathematics and Science Study (TIMSS) concluded, through its extensive research, that US students are not making the proper connections between new ideas and other meaningful concepts. This was due in large part because 60% of the time teachers do not help the students make these connections. In opposition to this is the fact that teachers in Japan, whose “students scored near the top throughout the study… made explicit connections in 96% of the lessons” (U.S. Department of Education (USDE). Pursuing Excellence, NCES 97-198. Washington, D.C.: National Center for Education Statistics, U.S. Government Printing Office, 1996, p.43). To summarize, there are a few nuggets to be found in this assignment: proper context, reiteration, help students make connections to previous knowledge, and stay current with the students interests. Implementing algebra is one of the hardest things I have seen in my class this year. The variables just throw the students off and they don’t know what to do with a letter. Students do fine as long as they have numbers but if you put letters (x, y, or z) they just lose focus. I have tried to explain to them that they need to take the variable and vision it as a number until they get there answer and then they have a real number to put in. I really liked how in “Building Conceptual Bridges” they took the chart and put the terms so that students could seem them side by side. This is a great tool for student to see and use until they were familiar enough with it that they didn’t need it. I still have students who ask what do I do with this letter. It is hard to take something and put it into math to make it more fun since there is a test that you have to have students pass. Being able to bring in a topic of diet in a country were obesity is becoming more prevalent is a great idea, and is a topic that you can use for more than just ratios and proportions. It blends in to graphs, and scatter plots, and from this you could bring in line of best fit, and slope. Being able to take data from a chart and putting it on a graph is great, and then being able to use it for more math, just makes that part of math more interesting. The lemonade mix is also a great problem, because what is better than to make lemonade on a hot day. The student could be split into two groups and each group makes a recipe and make enough for the whole class to try so that they can taste the difference between the two mixtures. This is something that the students would enjoy because it is different than the normal school day. These articles are great ideas of ways to make algebra more interesting and fun. Mitchell Reedy One thing I believe when teaching middle grades algebra is that we need to make a connection between real life and algebra. The students are old enough to understand and connect algebra to real life situations. I believe they would be able to understand the concepts better. Introducing and approaching algebra in middle grades, teachers need to have a very good understanding of it. The teacher needs to understand algebra and know different ways to teach it to accommodate one's students. All students learn differently, which means that different approaches need to be shown so that all students can understand what is trying to be taught. Of course you do not want to show 50 different ways to teach one concept but having an alternative if the first attempt doesn't work/click will be helpful. I agree with Mitchell about the "Building Conceptual Bridges" article the students were able to focus on what is being asked but in a chart view. This helps students get the picture of what is being done and once they get this concept they will be able to figure it out without the chart. Using a chart to help explain or differentiate what is being taught will help students understand materials. I would use that concept when teaching formulas for different algebraic concepts. Students need some way to help them organize the information they are being given. Danielle B.
As demonstrated by the article “Building Conceptual Bridges” by Joseph Martinez, many students in the United States do not understand basic algebraic reasoning even by the seventh and eighth grade. This may be because teachers only link new skills with previously learned concepts approximately 60% of the time. Middle school students, especially those who do not consider themselves strong in math, may experience anxiety at the mere mention of the word algebra. They may have a pre-conceived notion that algebra is difficult, entirely new, and useless. Removing the unknowns could also remove some of this anxiety. For example, when first introducing one-step equations, instead of using a variable, one could use an empty box, or a question mark to represent the unknown. When teaching students how to combine like terms, a number followed by a simple picture could represent a coefficient and a variable. One variable could be cookies, and one could be doughnuts, or healthier alternatives could be substituted.
The side-by-side comparison of arithmetic and algebraic expressions shown in the Martinez article would foster an awareness that the same rules and skills apply to numbers as with variables. In the “Lemonade Mix” problem, contributed by Francine Cabral Roy, a student may not even know that he or she is using algebra to complete the task. For example, first the students are to fill in a chart, which requires multiplying given quantities or completing a pattern. A student could complete each subsequent question a number of ways without ever realizing that he/she is using algebraic reasoning.
Application is the key. Students often benefit from a concrete connection to an abstract concept. Besides the lemonade problem, the article entitled “Diet, Ratios, Proportions: A Healthy Mix” by James A. Telese and Jesse Abete Jr. describes a seventh grade public school class that studied the correlation between fat, protein, and carbohydrates with fat content in popular school breakfast and lunch foods. The project encompassed numerous skills and lessons and continued throughout a sixth month period. I am impressed not only with the fact that one particular focus could maintain seventh grader’s interest for six months, but also how many different standards were covered with this single real-world connection. However, middle school students are general enthused by the mere mention of food. - Elizabeth Nahser Hugo Gareis As already discussed in this wiki, as well as in "Elementary and Middle School Mathematics (Van de Walle et al. 2010) and in the article "Building Conceptual Bridges (Martinez, 2002), one of the key issues in learning algebra is understanding the concept of "variable". As previously mentioned, the exercise referenced by Martinez, consisting of generalizing operations by first solving for numbers and then for variables and explaining the rational is very helpful in this regard. The other aspect of this, as also discussed previously in this wiki and exemplified very well in the article "Diet, Rations, Proportions: A Healthy Mix" (Telese & Abete, 2002), is t he impact of providing meaningful context to the introduction of algebraic concepts. In the lessons described by Telese and Abete, students observe the data regarding nutritional composition of common foods and then generalize with the use of variables; variables are thus introduced in a very natural way. Van De Walle also suggests the natural introduction of variables by substituting blank spaces or place boxes in verbally stated problems with letters, this is an easy to understand concept that can be utilized in early grades so that it will become natural to the students and eventually help them develop the ability to think in more abstract terms about variables. Another important point made by Van De Walle is that students should be made aware of the fact that variables can represent an unknown quantity or a quantity that can change. The curricular connections and the principle of building new concepts on top of concepts already mastered are extremely important as well. One very important connection that emerges from all the readings already referenced in this article is proportional reasoning; the connection between ratios, slopes, proportion problems and their generalization into algebraic constructs form an extremely useful and intuitive transition into algebra, this connection is certainly well exemplified in the Telese & Abete article. From the discussion presented by the authors of this wiki and by the referenced texts, the following main ideas emerge as strategies for introducing algebra concepts:
The purpose of algebra is to generalize mathematical operations and represent patterns
A meaningful, relevant context should be provided for the learning of algebraic concepts
It is important to build a good student grasp of the concept of variables
Patterns, tables and graphs are useful to help generalize arithmetic concepts
The use of relevant real life problems can not only enhance the engagement of students but also provide good vehicles to understand the symbolic/mathematics representation of the problems through variables and functions
It is crucial to provide the opportunity for students to use their existing knowledge to make sense of the new concepts before they are more formally (and algorithmically) introduced.
Perhaps one of the ways of describing the process of "doing algebra" might be: to observe, describe and solve a problem through the recognition of patterns and relationships, their generalization, and application to the problem at hand; using variables to represent the unknowns. I believe algebra can indeed be a lot of fun if it is presented in a relevant manner and if the students are giving the opportunity to discover the patterns and relationships and how they can be used to solve problems. I recently had a conversation with an adult who told me that he hated math but loved physics, which unfortunately is wholly dependent on math! I couldn't help to think what his attitude might be today if as a student he had been allowed to discover math in the context of his interest in physics.
I think Studnets need to see the connection of the real world in their classrooms, beacuse the student will be more engage in their learning. Ratios and proportion is a very important topic in understanding of algebra and higher mathemaics. In the artilce Diet, Ratio,Proportions: A Healthy Mix the teacher did a great job of engaging her student in learning algebra though what the students eat during lunch and breakfest. The student didnt really know that making a chart of the number of people coming to the fund rasier is consider algebra. That shows one exmaple how ratios and proportion could be use to teach algebra without studnets knowing it.
In buliding conceptual bridges, The author talks about how important it is to make connections between arthimetic and algebra. The worksheet that was shown throughout the article was very interesting to see the arthimetric and the algrebra on different sides. Joyce did touch base on this already, but if you look at joyce example above the aritmetic and the algebra is very simalir.
5+5+5+5= 4(5) is the same thing as saying 4(a) if those 5 were an A.
In the authors own algebra class, he was his studnets make connections using webs to relate old ideas to new ideas. I would like to use the web idea in my studnet teaching next fall.
This section is a repository of selected EDMG 519 Assignments and Postings for future reference.
Researching differences between reform and traditional instructional methods in mathematics
Classroom Participation
Assessment Instruments
Teaching Algebra
CRSP Assignment
Researching differences between reform and traditional instructional methods in mathematics (collaborative post)
The difference between traditional and reform instructional practices needs to begin with a history lesson. Our country has attempted to reform math instruction since after World War II. There were concerns the United States was lagging behind in technology due to the Russian launch of the satellite Sputnik. The space race had begun. However, prior to this launch the University of Illinois Committee on School Mathematics was already investigating problems with how high school mathematics was being taught. There were other influences involved in this “New Math” such as Piaget’s stages of cognitive development, Bruner’s discovery of mathematical ideas and his three stages of representation of mathematical ideas. New Math was in full swing in the 1960’s, however; in the 1970’s it all disappeared and teaching returned to “back to basics”. (Herrera,Owens, 2001, p.85)
The NCTM Standards-Based Reform approach to teaching mathematics was launched in 1989 after results from two international studies were published. They were the Second International Mathematics Study and the International Assessment of Educational Progress. Both studies demonstrated a decline or a stand-still of math scores in the United States. The National Commission for Excellence in Education published A Nation at Risk in 1983 sparking a sense of urgency with its quote, “If an unfriendly foreign power had attempted to impose on America the mediocre educational performance that exists today, we might have viewed it as an act of war.” (NCEE, 1983, p.5) There was also an update in 2000 Principles and Standards for School Mathematics.
The six principles of fundamental to high quality mathematics instruction from the Principles and Standards for School Mathematics are:
- The Equity Principle
- The Curriculum Principle
- The Teaching Principle
- The Learning Principle
- The Assessment Principle
- The Technology Principle
A traditional mathematics curriculum is based on direct instruction, such as, lecture style teaching, algorithms, and memorization. While a reform, or standards-based curriculum, is based on reasoning and construction through problem solving and investigation. (Baker, 2010)For Example, in a ttraditional mathematics curriculum, a student needs to know 3x5=15 before the student goes on to solve this word problem: Jones bought 3 different color pairs of socks and five different kinds of blue jeans. How many new combinations of socks and jeans can Jones make? (Latterell and Copes, 2003) Conversely, in a rreform mathematics curriculum, students learn the basics at the same time while solving far more practicle problems, so that students will be prepared to solve problems in real world situations. (Latterell and Copes, 2003) In tradition mathematics settings the students learn the basics so that they can explore and conquer more complex problems. In all aspects of learning you have to learn the basics before you can proceed to something harder. Critics may say that in some cases traditional instruction is less effective than other alternative methods of teaching but in some instances its the best for certain students/ group of students. Each part of the country/demographic area is different when it comes to learning. One area may learn best with traditional math instruction, where the school down the road may find a new mathematical method more effective.
Reform mathematics curriculum challenges student to make sense of new mathematical ideas through exploration and projects. Not by filling in a formula or doing the question/problem a certain way. Reform text books emphasize written and verbal communication, group work, making connections between concepts and representations, unlike traditional texts which emphasize step by step processes of problems with examples.
New Math has an emphasis on “ deductive reasoning, set theory, rigorous proof, and abstraction, while the Standards emphasize applications in the real world context, especially experimentation and data analysis.” (Herrera, Owens, (2001), p. 91) Critics of the standards-based approach suggest that reform mathematics puts too much emphasis on the effort, and not on the solution; educators are praising the process, regardless of the product. However, creators of standards-based curriculums argue that the specific content has not changed, “rather it shows how reasoning and sense making can be incorporated throughout the curriculum.” (Baker, 2010) The Standards-Based Reform has been slow to take off. Classes are still mostly lecture style with little conversation between teacher and students. Are we stubborn and do not want to give up our “right” or “wrong” approach to teaching and learning math? (Newton, 2007, p. 6) Or does government place too many demands on teachers and schools to implement the reform?
With the math reform, many new aspects of teaching math have came out. Instead of having students just sit and listen to a lecture during class, more group work and hands on studies are being introduced. At the school that I work at, we have started a new program this year, called Connected Math. This is a very hands on lesson and has more group work than individual work. With this program students are suppose to learn from each other instead of just the teacher. Van de Walle and Lovin both describe that group work is a better way for students to learn becasue they are learing from their peers instead or a superior in Teaching Student-Centered Mathematics.
References:
Baker, Linda. Numbers Wars: School Battles Heat Up Again in the Traditional versus Reform-Math Debate. Scientific American, March2010, 2p, 1 Illustration
Herrara, Terese A., Owens, Douglas T. The “New New Math”?: Two Reform Movements in Mathematical Education. Theory Into Practice, Spring2001, Vol. 40 Issue 2, p84, 9p
Latterell, Carman M., Copes Lawrence. "Can We Research Conclusion in Mathematics Education Research?. Phi Delta Kappani; Nov2003, Vol. 85 Issue 3, p207-211, 5p, 1 Illustration
Newton, Xiaoxia. Reflections on Math Reforms in the U.S.: A Cross-National Perspective. Education Digest, Sep2007, Vol. 73 Issue 1, p4-9, 6p
Lovin, LouAnn H., Van de Walle, John A. "Teaching Student-Centered Mathematics Grades 5-8", 2006, Vol.3, pg 1-37
Integration of Reform vs. Traditional Instructional Practices
“ Ten years after NCTM’s release of the Curriculum and Evaluation Standards for School Mathematics, this country is having a wrenching debate about what should be taught in mathematics and how it should be taught. Debate has degenerated to “math wars.” On one side are those who fervently believe children need to learn “the basics.” On the other side are those who believe or think they believe in the message of the Standards. These are the educators who believe in “reform mathematics.” Objectivity often gets lost in rhetoric and what either side believes is frequently vague at best (Van de Walle, 2003).”
Briefly, some advantages and disadvantages of each philosophy are listed below:
Would it not be possible then to instill the basics in students while actively seeking to utilize the concepts developed by the NCTM? In the text “Elementary and Middle School Mathematics: Teaching Developmentally (Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M., 2010, p. 5).”t t the authors refer to a document (The Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (2006)) created by experts who were brought together by the NCTM. Their thoughts on coherence basically suggest that the most significant concepts and skills be emphasized in each grade so as to form a fundamental core content for each grade.
With a more coherent structure in place that is built upon year after year, the teacher will have more time to focus on learning methods and spend less time attempting to connect the student’s present learning with learning from the past. The students will have a much better chance to make “connections” on their own. However, this is only one example of an approach that could be employed to integrate the two philosophies. In order to effectively integrate the two will require much work on the part of the teacher. The traditional methods will need to be studied, along with the reforms that have been suggested in an effort to align the two. It might be that this approach will need to be addressed with each individual lesson or concept. But, in order to address the most critical issue, the student’s education, such efforts are a necessity.
References
Van de Walle, J.A. (2003, April 1). Reform mathematics vs. the basics: understanding the conflict and dealing with it. Retrieved from http://mathematicallysane.com/reform-mathematics-vs-the-basics/
Interview with university of md mathematics professor jerome dancis (2006, February). [Electronic mailing list message]. Retrieved from http://www.baltimorecp.org/newsletter/BCPnews_feb06.htm#spotlight
Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2010). Elementary and middle school mathematics: teaching developmentally. Allyn & Bacon.
The comparison between traditional and reform practices in mathematics is marked by a healthy debate on the advantages of each educational approach. The authors of Elementary and Middle School Mathematics, point out that although empirical data regarding the effectiveness of each approach is still being gather, it appears clear that students in standard-based programs outperform traditional students in .problem-solving measures ( Van De Valle, Karp, Bay-Williams, 2010).
This page presents a collection of studies relevant to Mathematics reform.
- Learning From Teaching: Exploring the Relationship Between Reform Curriculum and Equity. Jo Boaler
The author discuses two studies which suggest that reform-based teaching and learning are central to the attainment of linguistic, ethnic, and class equity in schools.Journal for Research in Mathematics Education. July 2002, Volume 33, Issue 4, Pages 239 - 258
- Problem Solving as a Means Toward Mathematics for All: An Exploratory Look Through a Class Lens
Sarah Theule LubienskiJournal for Research in Mathematics Education. July 2000, Volume 31, Issue 4, Pages 454 - 482
Leubienski studied the experiences of 7th grade with problem-solving approaches to mathematics pedagogy.
- Selecting the Right Curriculum Brief http://www.nctm.org/news/content.aspx?id=12320
This brief from the National Council of Teachers of Mathematics analyzes the difficulty in producing conclusive comparative studies between traditional and standard-based methods of instruction. It does however suggests that the patterns of evidence indicate that while the both approaches appear to produce similar results in standardized computational tests, students taught with a standards-based approach seem "outperform students taught with traditional methods.Back to the top
Classroom Participation (collaborative observations from the EDMG 519 video "The Border Problem)
Role of the teacher:
· Facilitator
· Encourage different methods of solving the problem.
· Encourage students to demonstrate and explain to the rest of the class how they solved the problem.
· Encourage making connections between the different methods.
· Challenge the students to use one of the methods on a new problem.
- Provide feedback and praise.
- Bring into the discussion relevant information not brought up by the students.
- The teacher wrote down the different methods on the chalk board in expression format.
- Challange the students to find the last missing expressions from the board.
- Challange the students to explain not only how but why they used the chosen method.
- Maintain focus and interest throughout discussion.
v Thoroughly understand the material.v Elicit strategies from the students.
v Discuss the various strategies.
v Create a spirit of inquiry, trust, and expectation.
v Make lessons relevant to real world applications.
v Demonstrate various approaches of problem solving.
v Involve technology in the classroom.
v Elicit deeper thought as to method used.
v Guide discussions.
v Show respect for the students.
v Ask contemplative questions of the students.
- Challenge students/make them think/use what they know to solve abstract problems
- Make sure students understand material that is being taught
- Let students take turns answering and explaining their answers.
- Writing down notes/equations students come up with to solve specific problems
1. Have tools and manipulatives that put math into world context, so they can see how it will affect them.2. Earn the respect of the students by showing them that they care about their education, and by having the materials ready for the students.
3. Be open minded to new ideas, from other teachers and students.
4. Be constantly advancing in new technology in the class room.
Role of the students:
· Explain to the rest of the class how they solved the problem correctly.
· Explain to the rest of the class how they solved the problem incorrectly.
· Make sure they understand how the other students solved the problem correctly.
· Raise their hand quietly and wait to be called on (perhaps this one belongs in norms?)
- work productively with the member in their group
- participate in class discussion/ be ready to answer questions
- Listen to others
- Ask questions if you don't understand material
- Respect classmates and their thoughts.
- Participate in the discussion.
- Think of their own solution/opinion.
- Work in cooperation and community with others.
v Actively participate in classroom activities.v Learn from the methods of others within the class.
v Listen to others without them fearing other student’s derision.
v Work with others while demonstrating respect.
v Exhibit respect for the teacher.
v Learn.
1. Pay attention during class to the teacher and work.
2. Respect other student and teachers.
3. Keep up with text books, calculator, and notes from the class.
4. Contribute to the class every time therer is a discussion time.
5. Help other students understand what their method was in working a problem.
6. Have, learn proper knowledge of terms and wordings so explanations can be understood by entire class.
Norms of the classroom:
· Raise your hand to respond to the teacher’s request.
· Do not distract from the discussion.
Students know that when going to explain something in front of the class a pointer may be helpful.
Students showed that they needed to raise their hand and put it down if they were not called upon.
Students sat in their desks the right way and knew where they were suppose to sit.
Student's new to give their attention to whoever was speaking whether it was the teacher or a fellow classmate.
- Time management and control
- Participate in the discussion without interupting teacher or other students.
- Wait 3 minutes before raising hand to allow time for everyone to think about the question.
- Everyone must listen attentively and quietly when someone else is speaking.
- Faster and slower, more confident and less confident, students in the classroom.
v Teacher creates a spirit of inquiry, trust, and expectation.v Teacher involves technology in the classroom.
v Teacher guides discussions.
v Teacher shows respect for the students.
v Teacher asks contemplative questions of the students.
v Students actively participate in classroom activities.
v Students learn from the methods of others within the class.
v Students listen to others without them fearing other student’s derision.
v Students work with others while demonstrating respect.
v Students exhibit respect for the teacher.
Assessment Instruments
I agree that the P-PT tests skills rather than critical thinking. Also, the test writer made a few mistakes with the notation of a few of the problems (#20, 21, 23, & 24). The correct answer is not among the choices. For example #24; the question askes for 1.05 divided by 54.5. This yields 0.019 which is not among the choices. However, 54.5 divided by 1.05 yields 51.9, which is among the choices. (WB)
The pre-post test assesses basic number skills taught as "rules" in the math class.
Joyce, I agree[rfc]
I agree with Joyce that the pre-post test was seeing if the students knew the rules. I felt this test was testing students in standardize testing format ( multiple choice questions). (JT) I agree. (DB)
I total agree with you on that (Mitchell Reedy)
All of the questions on the Pre-Post Test are algorithmic, and involve decimals. There are no word problems or real-world examples except for the bonus question. The test is purely multiple choice, and the order of the questions suggests a specific sequence of skills. Often there are multiple questions testing a single skill presented in only one way. (EN)
The NAEP Item test assesses critical thinking skills learned in math class.
Joyce, I agree[rfc]
I agree with Joyce and I would use this test over the pre-post test because the critical thinking it involves. (JT)
I agree with JT & Joyce. (DB)
Our school had to take the NAEP test this year, and it was amazing how the calculator was not allowed in all parts, and that not all parts were multiple choice and they had to write out there answers and show all their work.
The NAEP test covers a variety of topics and skills, and each question is presented in a different way, even when accessing similar skills. There is a combination of multiple choice, short answer, discussion, and yes/no questions. Several of the questions could have more than one answer, or there are numerous pathways or methods that could be used to arrive at a solution. The test asks "Did you use a calculator?" after a few of the questions, implying that students may choose whether or not they would like to use one. (EN)
I like the how the NAEP is a variety of skills & topics and that the students really have to think about the question being asked instead of having just multiple choice. I think the students get to really show what they know with this test compare to P-PT. (DB)
Perhaps my strongest first impression was a huge contrast in the way I felt taking these tests. The P-PT immediately felt that it was difficult, tedious, and meaningless (just a bunch of annoying numbers); and I felt somewhat insecure about some of my answers even though I know the subject matter. In short: it was no fun at all!. In contrast, the NAEP felt like more fun, I felt more confident that even if I didn't know the answers or methods I could figure them out, and it was actually fun and challenging figuring them out. I felt more confident in my answers because the were relevant and make sense with respect to the evidence and clues I had in front of me.
What types of information do I gain from each?
I also agree. The NAEP test requires the students to understand the concepts involved and perhaps to assess how well they can visualize a problem. Also, this test requires the students to think more abstractly rather than just manipulating numbers. (WB)
PPT- The students' abilities to follow the basic rules of arithmetic.
Joyce, I agree[rfc]
The Pre-Post Test demonstrates the student's ability to reproduce algorithms previously taught in class with or without understanding, and with only short-term memorization. (EN
Will you truly know if your students can follow the rules of arithmetic from this test? I wouldn't know if my student knew the rules base on a multiple choice questions test. (JT)
I agree with you JT. You cannot really know what your students know by the P-PT. I think NAEP is a better way to assess that. (DB)
NAEP- The students' abilities to critically think about fraction and their relationships to one another.
Joyce, I agree[rfc]
I agree (Mitchell Reedy)
I agree with Joyce (JT)
The NAEP Test requires more critical thinking, and accesses a student's ability and understanding in applying previously taught skills to real-world situations. A students capability to solve a problem based on given examples is also tested. (EN) I agree (DB)
In addition of the good points already made, I think that as a teacher I gain a better insight into how well students understand the underlying ideas of the meaning of mathematics, of "doing mathematics"(De Valle, 2010). Also, I think the students themselves gain further understanding of the concept s being tested because they have to use their problem solving and critical thinking skills in completing the tasks, I am sure they would find many 'eureka" moments during the assesment that they might not have found before.
How might these items be an indicator of mathematics instruction in this classroom?
I think these tests will accurately reflect the type of instruction that the students receive on a daily basis. With the P-PT the students likely sit and work through problems out of te text each day without really understanding why they are doing this and the implications of the knowledge. The NAEP class will be more likely to have discussions concerning the concepts involved and will be required to work through real world applications that lead them to the fundamentals as a method of solving the problem. (WB)
I completely agree with you WB. Students need to understand the concepts they are learning so they can use them in everyday life and not just on tests.(DB)
PPT- More structured- straight from the textbook.
Joyce, I agree[rfc]
teaching towards a Standardize test (JT)
The Pre-Post Test class most likely uses lower-level demands, with explicit instruction techniques and no connection to underlying concepts. The focus is more on reproduction and correct answers than on understanding and cognitive development. (EN)
NAEP- Appropriate manipulatives were used and skills of drawing representations of the fractions were taught.
Joyce, I agree I would say however that it appears that skills of drawing representations were taught.[rfc]
My belief is the teacher will be having a discussion during the lesson with student's' to engage them in there learning. I also think the students will be working in groups to work with appropriate manipulatives were used and drawing representations of the problem to help solve the problem. (JT)
The NAEP Test class most likely uses higher level demands including procedures with connections and doing math. Skills are presented in a variety of ways, with the focus on developing a deeper understanding and either developing procedures, or applying previously learned material to real-world examples. (EN)
Yes, and I think the teaching strategy goes even beyond that to create in the students the ability and confidence that enables them to "do math". The NAEP test certainly seems to have built-in the expectation that students can use basic principles, together with their thinking and learning skills, to figure things out without the need of memorized algorithms if necessary.
I have another point here. If pre and post test activities from one instrument are desired, I think the PPT model will do a better job. Otherwise the NAEP test wi;ll do a better job of reflecting the student's abilities. [rfc]
My initial impression of the two assessments is that they are both totally different. The NAEP uses nothing but fractions for their questions, and the Pre-Post Test uses all decimals. The NAEP test also hits many different topics and the Pre-Post Test was addition, subtracting, and rounding. The NAEP test pushed for students to be able to work problems by hand instead of using the calculator for everything. The Pre-Post test didn’t push for one way or another, the use of calculator or not. They both are good tools because you get to see examples of what might be on the EOG’s, and you have the opportunity to teach students the importance of being able to do math without a calculator. With the Pre-Post Test, you get a sense of importance of estimating, which can sometimes can be over looked since students do it from 3rd grade up. Also a good number sense will help students with estimations, and being able to work without a calculator. These to samples, give a good insight as to some material that students will possible see again on a bigger test. Also it made me want to stress that students need to try to work problems without a calculator before they put it in there calculator. This can be an indicator of what happens in the classroom, just by the fact that one test says, no calculator, and the other doesn’t. One class has to use paper and pencil to solve problems. The other class seems like they might be more relaxed and just have to get close to the answer since it was more estimations. Both are good things to work on in a class, but they both need to be worked on.
CRSP Assignment
Power Point on Solid Geometry:
Teaching Algebra
The examples provided in the article Building Conceptual Bridges by Joseph G. R. Martinez are useful in helping middle school students transition from math to algebra.
2 + 2 + 2 = 6 or 3(2) a + a + a = 3a or 3(a)
22 ⋅ 23 = 32 or 25 a2 ⋅ a3 = a5
When introducing two step equations I have used candy to represent the variables and the constant.
I used Jolly Ranchers to show the variables and Hersey Kisses to show the constant. This was to show them they cannot mix the two.
I will use ∞ for Jolly Ranchers and Δ for Hersey Kisses .
3x + 2 = 8
∞∞∞ + ΔΔ = ΔΔΔΔΔΔΔΔ
-ΔΔ -ΔΔ
∞∞∞ = ΔΔΔΔΔΔ
We can only subtract Jolly Ranchers from Jolly Ranchers in this problem. The candy represents the two separate parts of the equation and they cannot simply mix the two.
This gets the students to a one step equation they are more familiar with.
This exercise and the related articles reiterate the fact that perhaps the best way to reach students with new mathematic concepts is to place the lessons into a format that holds some interest for them. The ideas of the lemonade mix, and also the dietary concerns of adolescents, both hit this target. Of course the problem of the lemonade mix is an older, more tried and true implementation of contextualizing to gain students interest. However, with the dietary concern project, there are several aspects that should be discussed.
First would be the use of the media to find topics and establish the parameters for projects. Dietary concern seems to be a great application that will pique the interest of the students. While this is not a topic I would have used 30 years ago, by keeping current with the students interests will lead to such topics. As stated in Diet, Ratios, Proportions: A Healthy Mix “the importance of choosing a context should not be overlooked” (p. 13).
Also, this is a great way to enter into conversations and learning concerning ratios. I have always found the use of ratios to be a very simple and effective way to answer the question “why do I need to know algebra?” Collectables, gas money, allowances, job opportunities are just a few.
Furthermore, through this type of instruction, the implementation of charts and graphs becomes easy, almost second nature. And, by setting this up as a longer term project it will give the students ample opportunity to learn and continue to delve deeper into this interdisciplinary topic.
The Third International Mathematics and Science Study (TIMSS) concluded, through its extensive research, that US students are not making the proper connections between new ideas and other meaningful concepts. This was due in large part because 60% of the time teachers do not help the students make these connections. In opposition to this is the fact that teachers in Japan, whose “students scored near the top throughout the study… made explicit connections in 96% of the lessons” (U.S. Department of Education (USDE). Pursuing Excellence, NCES 97-198. Washington, D.C.: National Center for Education Statistics, U.S. Government Printing Office, 1996, p.43).
To summarize, there are a few nuggets to be found in this assignment: proper context, reiteration, help students make connections to previous knowledge, and stay current with the students interests.
Implementing algebra is one of the hardest things I have seen in my class this year. The variables just throw the students off and they don’t know what to do with a letter. Students do fine as long as they have numbers but if you put letters (x, y, or z) they just lose focus. I have tried to explain to them that they need to take the variable and vision it as a number until they get there answer and then they have a real number to put in. I really liked how in “Building Conceptual Bridges” they took the chart and put the terms so that students could seem them side by side. This is a great tool for student to see and use until they were familiar enough with it that they didn’t need it. I still have students who ask what do I do with this letter. It is hard to take something and put it into math to make it more fun since there is a test that you have to have students pass. Being able to bring in a topic of diet in a country were obesity is becoming more prevalent is a great idea, and is a topic that you can use for more than just ratios and proportions. It blends in to graphs, and scatter plots, and from this you could bring in line of best fit, and slope. Being able to take data from a chart and putting it on a graph is great, and then being able to use it for more math, just makes that part of math more interesting. The lemonade mix is also a great problem, because what is better than to make lemonade on a hot day. The student could be split into two groups and each group makes a recipe and make enough for the whole class to try so that they can taste the difference between the two mixtures. This is something that the students would enjoy because it is different than the normal school day. These articles are great ideas of ways to make algebra more interesting and fun. Mitchell Reedy
One thing I believe when teaching middle grades algebra is that we need to make a connection between real life and algebra. The students are old enough to understand and connect algebra to real life situations. I believe they would be able to understand the concepts better. Introducing and approaching algebra in middle grades, teachers need to have a very good understanding of it. The teacher needs to understand algebra and know different ways to teach it to accommodate one's students. All students learn differently, which means that different approaches need to be shown so that all students can understand what is trying to be taught. Of course you do not want to show 50 different ways to teach one concept but having an alternative if the first attempt doesn't work/click will be helpful. I agree with Mitchell about the "Building Conceptual Bridges" article the students were able to focus on what is being asked but in a chart view. This helps students get the picture of what is being done and once they get this concept they will be able to figure it out without the chart. Using a chart to help explain or differentiate what is being taught will help students understand materials. I would use that concept when teaching formulas for different algebraic concepts. Students need some way to help them organize the information they are being given. Danielle B.
As demonstrated by the article “Building Conceptual Bridges” by Joseph Martinez, many students in the United States do not understand basic algebraic reasoning even by the seventh and eighth grade. This may be because teachers only link new skills with previously learned concepts approximately 60% of the time. Middle school students, especially those who do not consider themselves strong in math, may experience anxiety at the mere mention of the word algebra. They may have a pre-conceived notion that algebra is difficult, entirely new, and useless. Removing the unknowns could also remove some of this anxiety. For example, when first introducing one-step equations, instead of using a variable, one could use an empty box, or a question mark to represent the unknown. When teaching students how to combine like terms, a number followed by a simple picture could represent a coefficient and a variable. One variable could be cookies, and one could be doughnuts, or healthier alternatives could be substituted.
The side-by-side comparison of arithmetic and algebraic expressions shown in the Martinez article would foster an awareness that the same rules and skills apply to numbers as with variables. In the “Lemonade Mix” problem, contributed by Francine Cabral Roy, a student may not even know that he or she is using algebra to complete the task. For example, first the students are to fill in a chart, which requires multiplying given quantities or completing a pattern. A student could complete each subsequent question a number of ways without ever realizing that he/she is using algebraic reasoning.
Application is the key. Students often benefit from a concrete connection to an abstract concept. Besides the lemonade problem, the article entitled “Diet, Ratios, Proportions: A Healthy Mix” by James A. Telese and Jesse Abete Jr. describes a seventh grade public school class that studied the correlation between fat, protein, and carbohydrates with fat content in popular school breakfast and lunch foods. The project encompassed numerous skills and lessons and continued throughout a sixth month period. I am impressed not only with the fact that one particular focus could maintain seventh grader’s interest for six months, but also how many different standards were covered with this single real-world connection. However, middle school students are general enthused by the mere mention of food. - Elizabeth Nahser
Hugo Gareis
As already discussed in this wiki, as well as in "Elementary and Middle School Mathematics (Van de Walle et al. 2010) and in the article "Building Conceptual Bridges (Martinez, 2002), one of the key issues in learning algebra is understanding the concept of "variable". As previously mentioned, the exercise referenced by Martinez, consisting of generalizing operations by first solving for numbers and then for variables and explaining the rational is very helpful in this regard. The other aspect of this, as also discussed previously in this wiki and exemplified very well in the article "Diet, Rations, Proportions: A Healthy Mix" (Telese & Abete, 2002), is t he impact of providing meaningful context to the introduction of algebraic concepts. In the lessons described by Telese and Abete, students observe the data regarding nutritional composition of common foods and then generalize with the use of variables; variables are thus introduced in a very natural way. Van De Walle also suggests the natural introduction of variables by substituting blank spaces or place boxes in verbally stated problems with letters, this is an easy to understand concept that can be utilized in early grades so that it will become natural to the students and eventually help them develop the ability to think in more abstract terms about variables. Another important point made by Van De Walle is that students should be made aware of the fact that variables can represent an unknown quantity or a quantity that can change. The curricular connections and the principle of building new concepts on top of concepts already mastered are extremely important as well. One very important connection that emerges from all the readings already referenced in this article is proportional reasoning; the connection between ratios, slopes, proportion problems and their generalization into algebraic constructs form an extremely useful and intuitive transition into algebra, this connection is certainly well exemplified in the Telese & Abete article.
From the discussion presented by the authors of this wiki and by the referenced texts, the following main ideas emerge as strategies for introducing algebra concepts:
- The purpose of algebra is to generalize mathematical operations and represent patterns
- A meaningful, relevant context should be provided for the learning of algebraic concepts
- It is important to build a good student grasp of the concept of variables
- Patterns, tables and graphs are useful to help generalize arithmetic concepts
- The use of relevant real life problems can not only enhance the engagement of students but also provide good vehicles to understand the symbolic/mathematics representation of the problems through variables and functions
- It is crucial to provide the opportunity for students to use their existing knowledge to make sense of the new concepts before they are more formally (and algorithmically) introduced.
Perhaps one of the ways of describing the process of "doing algebra" might be: to observe, describe and solve a problem through the recognition of patterns and relationships, their generalization, and application to the problem at hand; using variables to represent the unknowns. I believe algebra can indeed be a lot of fun if it is presented in a relevant manner and if the students are giving the opportunity to discover the patterns and relationships and how they can be used to solve problems. I recently had a conversation with an adult who told me that he hated math but loved physics, which unfortunately is wholly dependent on math! I couldn't help to think what his attitude might be today if as a student he had been allowed to discover math in the context of his interest in physics.I think Studnets need to see the connection of the real world in their classrooms, beacuse the student will be more engage in their learning. Ratios and proportion is a very important topic in understanding of algebra and higher mathemaics. In the artilce Diet, Ratio,Proportions: A Healthy Mix the teacher did a great job of engaging her student in learning algebra though what the students eat during lunch and breakfest. The student didnt really know that making a chart of the number of people coming to the fund rasier is consider algebra. That shows one exmaple how ratios and proportion could be use to teach algebra without studnets knowing it.
In buliding conceptual bridges, The author talks about how important it is to make connections between arthimetic and algebra. The worksheet that was shown throughout the article was very interesting to see the arthimetric and the algrebra on different sides. Joyce did touch base on this already, but if you look at joyce example above the aritmetic and the algebra is very simalir.
5+5+5+5= 4(5) is the same thing as saying 4(a) if those 5 were an A.
In the authors own algebra class, he was his studnets make connections using webs to relate old ideas to new ideas. I would like to use the web idea in my studnet teaching next fall.